3.183 \(\int \frac{\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=291 \[ \frac{429 a^9 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{32768 b^{15/2}}-\frac{429 a^8 \sqrt{a x+b x^{2/3}}}{32768 b^7 x^{2/3}}+\frac{143 a^7 \sqrt{a x+b x^{2/3}}}{16384 b^6 x}-\frac{143 a^6 \sqrt{a x+b x^{2/3}}}{20480 b^5 x^{4/3}}+\frac{429 a^5 \sqrt{a x+b x^{2/3}}}{71680 b^4 x^{5/3}}-\frac{143 a^4 \sqrt{a x+b x^{2/3}}}{26880 b^3 x^2}+\frac{13 a^3 \sqrt{a x+b x^{2/3}}}{2688 b^2 x^{7/3}}-\frac{a^2 \sqrt{a x+b x^{2/3}}}{224 b x^{8/3}}-\frac{\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}-\frac{a \sqrt{a x+b x^{2/3}}}{16 x^3} \]

[Out]

-(a*Sqrt[b*x^(2/3) + a*x])/(16*x^3) - (a^2*Sqrt[b*x^(2/3) + a*x])/(224*b*x^(8/3)
) + (13*a^3*Sqrt[b*x^(2/3) + a*x])/(2688*b^2*x^(7/3)) - (143*a^4*Sqrt[b*x^(2/3)
+ a*x])/(26880*b^3*x^2) + (429*a^5*Sqrt[b*x^(2/3) + a*x])/(71680*b^4*x^(5/3)) -
(143*a^6*Sqrt[b*x^(2/3) + a*x])/(20480*b^5*x^(4/3)) + (143*a^7*Sqrt[b*x^(2/3) +
a*x])/(16384*b^6*x) - (429*a^8*Sqrt[b*x^(2/3) + a*x])/(32768*b^7*x^(2/3)) - (b*x
^(2/3) + a*x)^(3/2)/(3*x^4) + (429*a^9*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3)
+ a*x]])/(32768*b^(15/2))

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Rubi [A]  time = 0.882181, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{429 a^9 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{32768 b^{15/2}}-\frac{429 a^8 \sqrt{a x+b x^{2/3}}}{32768 b^7 x^{2/3}}+\frac{143 a^7 \sqrt{a x+b x^{2/3}}}{16384 b^6 x}-\frac{143 a^6 \sqrt{a x+b x^{2/3}}}{20480 b^5 x^{4/3}}+\frac{429 a^5 \sqrt{a x+b x^{2/3}}}{71680 b^4 x^{5/3}}-\frac{143 a^4 \sqrt{a x+b x^{2/3}}}{26880 b^3 x^2}+\frac{13 a^3 \sqrt{a x+b x^{2/3}}}{2688 b^2 x^{7/3}}-\frac{a^2 \sqrt{a x+b x^{2/3}}}{224 b x^{8/3}}-\frac{\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}-\frac{a \sqrt{a x+b x^{2/3}}}{16 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^(2/3) + a*x)^(3/2)/x^5,x]

[Out]

-(a*Sqrt[b*x^(2/3) + a*x])/(16*x^3) - (a^2*Sqrt[b*x^(2/3) + a*x])/(224*b*x^(8/3)
) + (13*a^3*Sqrt[b*x^(2/3) + a*x])/(2688*b^2*x^(7/3)) - (143*a^4*Sqrt[b*x^(2/3)
+ a*x])/(26880*b^3*x^2) + (429*a^5*Sqrt[b*x^(2/3) + a*x])/(71680*b^4*x^(5/3)) -
(143*a^6*Sqrt[b*x^(2/3) + a*x])/(20480*b^5*x^(4/3)) + (143*a^7*Sqrt[b*x^(2/3) +
a*x])/(16384*b^6*x) - (429*a^8*Sqrt[b*x^(2/3) + a*x])/(32768*b^7*x^(2/3)) - (b*x
^(2/3) + a*x)^(3/2)/(3*x^4) + (429*a^9*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3)
+ a*x]])/(32768*b^(15/2))

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Rubi in Sympy [A]  time = 83.0062, size = 269, normalized size = 0.92 \[ \frac{429 a^{9} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x + b x^{\frac{2}{3}}}} \right )}}{32768 b^{\frac{15}{2}}} - \frac{429 a^{8} \sqrt{a x + b x^{\frac{2}{3}}}}{32768 b^{7} x^{\frac{2}{3}}} + \frac{143 a^{7} \sqrt{a x + b x^{\frac{2}{3}}}}{16384 b^{6} x} - \frac{143 a^{6} \sqrt{a x + b x^{\frac{2}{3}}}}{20480 b^{5} x^{\frac{4}{3}}} + \frac{429 a^{5} \sqrt{a x + b x^{\frac{2}{3}}}}{71680 b^{4} x^{\frac{5}{3}}} - \frac{143 a^{4} \sqrt{a x + b x^{\frac{2}{3}}}}{26880 b^{3} x^{2}} + \frac{13 a^{3} \sqrt{a x + b x^{\frac{2}{3}}}}{2688 b^{2} x^{\frac{7}{3}}} - \frac{a^{2} \sqrt{a x + b x^{\frac{2}{3}}}}{224 b x^{\frac{8}{3}}} - \frac{a \sqrt{a x + b x^{\frac{2}{3}}}}{16 x^{3}} - \frac{\left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{3 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**(2/3)+a*x)**(3/2)/x**5,x)

[Out]

429*a**9*atanh(sqrt(b)*x**(1/3)/sqrt(a*x + b*x**(2/3)))/(32768*b**(15/2)) - 429*
a**8*sqrt(a*x + b*x**(2/3))/(32768*b**7*x**(2/3)) + 143*a**7*sqrt(a*x + b*x**(2/
3))/(16384*b**6*x) - 143*a**6*sqrt(a*x + b*x**(2/3))/(20480*b**5*x**(4/3)) + 429
*a**5*sqrt(a*x + b*x**(2/3))/(71680*b**4*x**(5/3)) - 143*a**4*sqrt(a*x + b*x**(2
/3))/(26880*b**3*x**2) + 13*a**3*sqrt(a*x + b*x**(2/3))/(2688*b**2*x**(7/3)) - a
**2*sqrt(a*x + b*x**(2/3))/(224*b*x**(8/3)) - a*sqrt(a*x + b*x**(2/3))/(16*x**3)
- (a*x + b*x**(2/3))**(3/2)/(3*x**4)

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Mathematica [A]  time = 0.298075, size = 164, normalized size = 0.56 \[ \frac{429 a^9 \tanh ^{-1}\left (\frac{\sqrt{a x+b x^{2/3}}}{\sqrt{b} \sqrt [3]{x}}\right )}{32768 b^{15/2}}-\frac{\sqrt{a x+b x^{2/3}} \left (45045 a^8 x^{8/3}-30030 a^7 b x^{7/3}+24024 a^6 b^2 x^2-20592 a^5 b^3 x^{5/3}+18304 a^4 b^4 x^{4/3}-16640 a^3 b^5 x+15360 a^2 b^6 x^{2/3}+1361920 a b^7 \sqrt [3]{x}+1146880 b^8\right )}{3440640 b^7 x^{10/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^(2/3) + a*x)^(3/2)/x^5,x]

[Out]

-(Sqrt[b*x^(2/3) + a*x]*(1146880*b^8 + 1361920*a*b^7*x^(1/3) + 15360*a^2*b^6*x^(
2/3) - 16640*a^3*b^5*x + 18304*a^4*b^4*x^(4/3) - 20592*a^5*b^3*x^(5/3) + 24024*a
^6*b^2*x^2 - 30030*a^7*b*x^(7/3) + 45045*a^8*x^(8/3)))/(3440640*b^7*x^(10/3)) +
(429*a^9*ArcTanh[Sqrt[b*x^(2/3) + a*x]/(Sqrt[b]*x^(1/3))])/(32768*b^(15/2))

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Maple [A]  time = 0.022, size = 181, normalized size = 0.6 \[ -{\frac{1}{3440640\,{x}^{4}} \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{{\frac{3}{2}}} \left ( 45045\, \left ( b+a\sqrt [3]{x} \right ) ^{17/2}{b}^{15/2}-390390\, \left ( b+a\sqrt [3]{x} \right ) ^{15/2}{b}^{17/2}+1495494\, \left ( b+a\sqrt [3]{x} \right ) ^{13/2}{b}^{19/2}-3317886\, \left ( b+a\sqrt [3]{x} \right ) ^{11/2}{b}^{21/2}+4685824\, \left ( b+a\sqrt [3]{x} \right ) ^{9/2}{b}^{23/2}-4349826\, \left ( b+a\sqrt [3]{x} \right ) ^{7/2}{b}^{{\frac{25}{2}}}+2633274\, \left ( b+a\sqrt [3]{x} \right ) ^{5/2}{b}^{{\frac{27}{2}}}+390390\, \left ( b+a\sqrt [3]{x} \right ) ^{3/2}{b}^{{\frac{29}{2}}}-45045\,\sqrt{b+a\sqrt [3]{x}}{b}^{{\frac{31}{2}}}-45045\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){b}^{7}{a}^{9}{x}^{3} \right ) \left ( b+a\sqrt [3]{x} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{29}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^(2/3)+a*x)^(3/2)/x^5,x)

[Out]

-1/3440640*(b*x^(2/3)+a*x)^(3/2)*(45045*(b+a*x^(1/3))^(17/2)*b^(15/2)-390390*(b+
a*x^(1/3))^(15/2)*b^(17/2)+1495494*(b+a*x^(1/3))^(13/2)*b^(19/2)-3317886*(b+a*x^
(1/3))^(11/2)*b^(21/2)+4685824*(b+a*x^(1/3))^(9/2)*b^(23/2)-4349826*(b+a*x^(1/3)
)^(7/2)*b^(25/2)+2633274*(b+a*x^(1/3))^(5/2)*b^(27/2)+390390*(b+a*x^(1/3))^(3/2)
*b^(29/2)-45045*(b+a*x^(1/3))^(1/2)*b^(31/2)-45045*arctanh((b+a*x^(1/3))^(1/2)/b
^(1/2))*b^7*a^9*x^3)/x^4/(b+a*x^(1/3))^(3/2)/b^(29/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b*x^(2/3))^(3/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b*x^(2/3))^(3/2)/x^5,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**(2/3)+a*x)**(3/2)/x**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.4215, size = 316, normalized size = 1.09 \[ -\frac{\frac{45045 \, a^{10} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right ){\rm sign}\left (x^{\frac{1}{3}}\right )}{\sqrt{-b} b^{7}} + \frac{45045 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{10}{\rm sign}\left (x^{\frac{1}{3}}\right ) - 390390 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{10} b{\rm sign}\left (x^{\frac{1}{3}}\right ) + 1495494 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{10} b^{2}{\rm sign}\left (x^{\frac{1}{3}}\right ) - 3317886 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{10} b^{3}{\rm sign}\left (x^{\frac{1}{3}}\right ) + 4685824 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{10} b^{4}{\rm sign}\left (x^{\frac{1}{3}}\right ) - 4349826 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{10} b^{5}{\rm sign}\left (x^{\frac{1}{3}}\right ) + 2633274 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{10} b^{6}{\rm sign}\left (x^{\frac{1}{3}}\right ) + 390390 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{10} b^{7}{\rm sign}\left (x^{\frac{1}{3}}\right ) - 45045 \, \sqrt{a x^{\frac{1}{3}} + b} a^{10} b^{8}{\rm sign}\left (x^{\frac{1}{3}}\right )}{a^{9} b^{7} x^{3}}}{3440640 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b*x^(2/3))^(3/2)/x^5,x, algorithm="giac")

[Out]

-1/3440640*(45045*a^10*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))*sign(x^(1/3))/(sqrt(
-b)*b^7) + (45045*(a*x^(1/3) + b)^(17/2)*a^10*sign(x^(1/3)) - 390390*(a*x^(1/3)
+ b)^(15/2)*a^10*b*sign(x^(1/3)) + 1495494*(a*x^(1/3) + b)^(13/2)*a^10*b^2*sign(
x^(1/3)) - 3317886*(a*x^(1/3) + b)^(11/2)*a^10*b^3*sign(x^(1/3)) + 4685824*(a*x^
(1/3) + b)^(9/2)*a^10*b^4*sign(x^(1/3)) - 4349826*(a*x^(1/3) + b)^(7/2)*a^10*b^5
*sign(x^(1/3)) + 2633274*(a*x^(1/3) + b)^(5/2)*a^10*b^6*sign(x^(1/3)) + 390390*(
a*x^(1/3) + b)^(3/2)*a^10*b^7*sign(x^(1/3)) - 45045*sqrt(a*x^(1/3) + b)*a^10*b^8
*sign(x^(1/3)))/(a^9*b^7*x^3))/a